Mobility — where it appears
Named by 14 essays across 3 fields — each of them below, with the objects they name alongside it.
The margin a count needs
Leaving the opponent fewest replies names only surviving options nine times in ten, which leaves the question of what a bound stated in that count would have to be weakened to. It is a margin. Over 57,879 pairs of Domineering options, the one leaving the opponent fewer replies is the worse of the two 1,052 times at a margin of one and 72 times at a margin of two — and at a margin of three, never.
The weight that blunts the count
The rung below found that counting the opponent's replies gets the direction of a comparison right once the gap reaches three, and proposed a repair: weigh each reply by whether it leaves the opponent anything. Weighing it makes the count worse. The threshold goes from three to four, the failures from 1,124 to 1,320, and all seventy-two of the pairs the repair was written for come through it unchanged.
The threshold was a fact about the census
Two rungs failed to account for the seventy-two pairs where a mobility count gets the direction of a comparison wrong, and the third looks at them one at a time. They are not a class of shapes. All seventy-two are on the largest board in the census, at two depths, and sixteen positions up to symmetry — and one board larger the count fails at a margin of three, which the ladder has been quoting as the point at which it never does.
Three rules and a tie-break
An exhaustive table of what a Domineering strategy has to remember is 3,308 lines. Three rules applied in order answer 94.5 per cent of it — leave the opponent fewest replies, then keep the region whole, then take whichever placement comes first — and the fourth and fifth rules answer not one more. The residue is 181 decisions in which every rule scores the candidates the same and one of them is worse.
A threshold is a detection limit
The rung below had two points — a margin of three at fifteen squares, four at eighteen — and asked whether the mobility rule's threshold grows with the board. Eleven more sweeps say no property of a board orders the thresholds, that the same board at two depths gives two of them, and that a tenth of the sweep which produced the four reports three instead. What does move, on every board measured twice, is the depth.
Room pulls two ways
The rung below found the distance between two amazons setting a shared region's temperature and asked for something finer — the squares each can reach, or the squares both can. Neither beats the distance on its own. Together they beat it by half as much again, and the reason is that they pull opposite ways: further apart is hotter, and sharing more reachable squares is colder.
A heuristic that becomes a theorem
The mobility rule's failure rate had been measured at two depths on each of five boards and found to fall. Swept at every depth it does not merely fall — it accelerates, and it reaches exactly nought before the endgame. From four to eight empty squares onwards the rule has no exceptions at all, which turns a rule of thumb into a guarantee for the last few moves.
The price of taking the maximum
The seventy-two decisions where a Domineering strategy's rules name the wrong placement are never wrong by more than one reply, and a third of them are the second rule's fault rather than the mobility count's. The repair that follows — keep every placement within one reply of the best — retains a best placement every time and costs thirty decisions for every one it saves.
An effect that changes sign
Which squares two Amazons share turns out to matter about as much as how many — three shared squares in a line run at 0.63 where three scattered run at 2.51. But the effect of clumping is hotter at one distance and colder at the next, so the arrangement predicts well and describes nothing, which is not what the four rungs below it produced.
A catalogue that builds itself
A solver that stores every region it has to evaluate builds a catalogue out of its own games. After 650 games it holds 232 of the 1,042 shapes and is still growing — and the order things arrive in is nearly arbitrary while the order they are consulted in reproduces a census of a strong player's games almost exactly.
The third point on the curve
Four rungs below this one measure a shared Amazons region's temperature against how far apart its two amazons are, and every one does it on a 3 × 3 board, where the distance can only be 1 or 2. Two points make a direction, not a curve. A 4 × 4 board reaches distance 3 — and cannot be evaluated at all until the regions are cut to five free squares. Restricted that far, the answer is neither a sign that flips nor an oscillation: the rise continues and it is running out, the second step being 36 per cent of the first.
Eleven moves and one decision
A prefix has one quantifier a turn, so a game of eleven moves is eleven alternations. Counted on the boards themselves, a Toads and Frogs strip of eleven moves has twenty-six turns with exactly one move available and one turn anywhere at which the choice changes the answer; a Clobber board has a hundred and fourteen turns and none. Nim, the game everybody calls solved, decides at four turns in five.
The opponent stops choosing
Replace one player by a rule with no search in it and the question has one chooser left, which is a puzzle rather than a game. Nim recovers five of its six lost positions that way, and six of seven on three heaps of five. Domineering recovers six of a hundred and twenty-two while the fixed rule throws away a winning move eighty-eight times, and one Clobber board recovers none at all — because on that board no rule can misplay.
A turn is not a bit
The prefix a game is read as gives each player one quantifier a turn, and a turn on a board is a choice among however many moves there are. Nim on heaps of 3, 4 and 5 lasts twelve moves and carries 23.6 bits of choice; a Toads and Frogs strip lasts eleven and carries two. Corrected for that, the model predicts a strategy 539 times too large on one board and 67 times too small on another, and the two failures have different causes.
Named alongside it
The objects these essays reach for when they reach for this one.
HeuristicApproximationEnumerationDomineeringCounterexampleValueDecompositionBoundComparisonDominanceExhaustive searchRegion